A theory of swept hinge lines

This page is currently under construction, last updated April 24 2014. Stay tuned for more! Some of these ideas could be conveyed by sketches and captions better than by words and photos, so I hope to add those items in the future.

This page is meant to supplement the related page "Notes on sail billow, dihedral, and anhedral in flex-wing hang gliders". We'll assume readers are familiar with that content, and particularly with the concept of assessing the dihedral or anhedral geometry of an aircraft by looking at the aircraft from the side. In "Notes on sail billow, dihedral, and anhedral in flex-wing hang gliders", we saw that sail billow creates dihedral in the inboard portions of a wing and anhedral in the outboard portions of the wing. One way to conceptualize these relationships is via a "theory of swept hinge lines".

Consider a matched pair of flaps or elevons located at the trailing edges of an aircraft's left and right wings. Consider that the trailing edges of the wings are swept aft, and the hinge line of the flaps or elevons are also swept aft. If we lift the flaps or elevons upwards (raising their trailing edges), we create an anhedral geometry. Looking from the side, we can see the top surface of the flap or elevon nearest the camera and the bottom surface of the flap or elevon furthest from the camera. (Photo to be added.) If the relative wind or airflow is blowing sideways across the aircraft, it will tend to strike the top surface of the "upwind" flap or elevon and the bottom surface of the "downwind" flap or elevon, creating a roll torque in the "upwind" direction.

Here are some additional photos that show this geometry. (Photos to be added.)

Now consider what happens when we deflect these same surfaces downward. Looking from the side, we can now see the bottom surface of the flap or elevon nearest the camera and the top surface of the flap or elevon furthest from the camera. (Photo to be added.) This is a dihedral geometry. If the relative wind or airflow is blowing sideways across the aircraft, it will tend to strike the bottom surface of the "upwind" flap or elevon and the top surface of the "downwind" flap or elevon, creating a roll torque in the "downwind" direction.

Here are some additional photos that show this geometry: ATOS with flaps deployed, Aerianne Swift with flaps deployed. (In both photos, the bottom of surface of the left flap is clearly visible. The top surface of the right flap is also visible, slightly obscured by the sun glinting off of it.)

We can state a comprehensive "theory of swept hinge lines" as follows:

When we have a matched pair of surfaces or panels on an aircraft's left and right wings, and the surfaces are hinged on their leading edges, then raising both surfaces upwards will have the following result: we'll create an anhedral geometry if the hinge lines are swept aft, and we'll create a dihedral geometry if the hinge lines are swept forward. The more the sweep in the hinge lines, the stronger the anhedral or dihedral geometry result from a given angular deflection of the surfaces. If the hinge lines are parallel to the aircraft's longitudinal axis, as viewed from above, then the sweep angle is of the hinge lines is considered to be 90 degrees-- this is the most extreme case. In this case, if the moving surfaces are hinged on their inboard edges, the hinge lines are considered to be swept forward-- raising the surfaces will create a dihedral geometry. On the other hand, if the hinge lines are parallel to the aircraft's longitudinal axis as viewed from above and the moving surfaces are hinged on their outboard edges, then the hinge lines are considered to be swept aft-- raising the surfaces will create an anhedral geometry.

Conversely, when we have a matched pair of surfaces or panels on an aircraft's left and right wings, and the surfaces are hinged on their leading edges, then lowering both surfaces downwards will have the following result: we'll create an dihedral geometry if the hinge lines are swept aft, and we'll create an anhedral geometry if the hinge lines are swept forward. Again, the more the sweep in the hinge lines, the stronger the anhedral or dihedral geometry result from a given angular deflection of the surfaces. If the hinge lines are parallel to the aircraft's longitudinal axis, as viewed from above, then the sweep angle is of the hinge lines is considered to be 90 degrees-- this is the most extreme case. In this case, if the moving surfaces are hinged on their inboard edges, the hinge lines are considered to be swept forward-- lowering the surfaces will create an anhedral geometry. On the other hand, if the hinge lines are parallel to the aircraft's longitudinal axis as viewed from above and the moving surfaces are hinged on their outboard edges, then the hinge lines are considered to be swept aft-- lowering the surfaces will create a dihedral geometry.

These effects are undoubtedly well known-- moving panels with swept hinge lines are very common on airplanes. Consider the moving control surfaces on the trailing edges of a highly swept wing, or on the trailing edges of a tapered wing with a straight leading edge and swept-forward trailing edge. We're not saying that the net effect of the deflecting these panels in the presence of a sideways flow is always easy to predict-- vortices shed off of the discontinuities in the structure may contribute roll torques in ways that are not obvious at first glance. We're just saying that if an undisturbed free airstream has a sideways component, it will "see" an anhedral or dihedral geometry as it encounters these deflected surfaces.

Here's one unusual way to mount a moving control surface on a swept-wing aircraft without having a swept hinge line: look at the ailerons at the extreme aft ends of the wing of this English Electric "Lightning" interceptor. The hinge lines of the ailerons are not swept. The hinge lines of the downward-deflected flaps on the inboard parts of the wing are strongly swept, however, and we can see the bottom side of the flap on the wing nearest the camera and the top side of the wing furthest from the camera. This creates a dihedral geometry-- a sideways airflow would tend to increase the lift generated by the flap on the "upwind" wing and decrease the lift generated by the flap on the "downwind" wing, generating a "downwind" roll torque.

We can also use the "theory of swept hinge lines" to predict the anhedral or dihedral geometry that will be created by rotating the entire left and right wings around each of their leading edges.

This metal model (photo to be added) has two "leading edge tubes" that are fixed in the same plane as the "keel tube" or "fuselage". However the "wings" can be rotated around the leading edge tubes, either by lifting or lowering their trailing edges. In this photo (model figure 1) we've raised the trailing edges of the wings. Now we've created an anhedral geometry-- in the photo, we can see the top surface of the wing nearest the camera, while we have an edge-on view of the wing furthest from the camera. During a sideslip, this geometry will create an "upwind" roll torque.

Here is another view (model figure 2) of this configuration. The model still has the same geometry, but we've rotated the whole model in a nose-up fashion so that the chord line of the wing is horizontal. Now the anhedral geometry is obvious. In this view, we can see the top surface of the wing nearest the camera and the bottom surface of the wing furthest from the camera.

Note that the "leading edge tubes" are still in plane with the "keel tube", but the leading edge tubes now have a downward droop relative to chord line of the wing. When we rotated each wing panel by lifting its trailing edge, using the swept leading-edge tubes as the hinge lines, we fundamentally changed the "aircraft's" geometry.

Here's a head-on view (model figure 3) of the model in this configuration, with the camera "in plane" with the mean chord line of each wing.

In this photo (model figure 4), we've taken the same metal model and rotated the wing panels in the opposite direction, lowering the trailing edges. Again, the swept leading-edge tubes are serving as the hinge lines, and the leading-edge tubes are remaining "in plane" with the keel tube. Now we've created a dihedral geometry. In the photo, we can see the bottom surface of the wing nearest the camera, and we can see the top surface of the wing furthest from the camera. During a sideslip, the "upwind" wing will have a higher angle-of-attack than the "downwind" wing, and this will create a "downwind" roll torque.

Here's a head-on view (model figure 5) of the model in this configuration, with the camera "in plane" with the mean chord line of each wing. The dihedral geometry is clear-- when the mean chord line of each wing is horizontal, the leading edges are "bent" upwards.

If these leading-edge tubes were swept forward rather than aft, all these effects would be reversed. If the leading-edges were not swept at all, then rotating the wing panels around the leading-edges would not create any anhedral or dihedral geometry at all.

The "theory of swept hinge lines" that we formulated to explain the geometry of moving flaps and elevons, also completely explains the geometry that we're seeing with this metal model where the entire wing panels are rotating around the fixed leading-edge tubes.

Now consider the effect of sail billow and washout in a flex-wing hang glider. How is the wing shape different than it would be if the sail were stretched tightly between the leading-edge tubes and keel tubes, with no billow or washout? It's tempting to visualize each wing panel as being somewhat rotated around the leading-edge tube, as if the entire trailing edge of each wing were lifted upwards by billow and washout, as in the above photos of the metal model with anhedral. Certainly that's what happens to the mean chord line-- the more the sail billow and washout, the more the mean chord line of the wing is rotated in a trailing-edge-up, leading-edge-down direction, compared to the keel tube. If we try to quantify the wing's anhedral by measuring the downward droop in the leading edge tubes with the aircraft positioned sin pitch attitude where the mean chord line is horizontal, we'll certainly measure more anhedral when the sail is highly billowed and washed out, than when it is not.

That's more or less what we've done in these photos. (Photo 1: VG on. Photo 2: VG off.) The glider is an Airborne Blade with a pulley VG system. The glider is standing on its kingpost, inverted, to tension the lower rigging with the weight of the sail. Rather than calculate the "mean chord line" of the wing, we've simply positioned the glider so that the mid-span chord line is horizontal-- see how the trailing edge disappears behind the leading edge at the mid-span of each wing in both photos. In the first photo, the VG is tight. In the second photo, we've loosened the VG, increasing sail billow and washout and rotating the mid-span chord line in the trailing-edge-up, leading-edge-down direction. To re-level the mid-span chord line, we've pitched the nose of the glider up. This happens in actual flight-- we feel the control bar come forward as we slack the VG. This nose-up rotation of the whole airframe has caused the swept leading edge tubes to have more down-droop when viewed from the front-- i.e. more anhedral relative to the horizon line. There's more vertical distance between the outboard tips of the leading edge tubes and the inboard ends of the leading-edge tubes than when the VG was tight. Likewise there's more vertical distance between the quarter-chord points near the wingtips and the quarter-chord point of the wing root than when the VG was tight. The effect would be even more pronounced in a glider with a cam VG-- with a cam VG, there wouldn't be any decrease in "airframe anhedral"-- the downdroop of the leading edge tubes in relation to the keel tube-- when we loosen the VG.

We could argue that this is a pretty good depiction of how tightening the VG decreases anhedral, and loosening the VG increases anhedral. However, the truth is that this really isn't a very good "holistic" look at the complex three-dimensional shape of a flex-wing hang glider. It would be totally appropriate to focus on the mean chord line (or the mid-span chord line) in this manner if the left and right wing panels were each simply rotating around their leading-edge tubes, as illustrated in figures 1-3 with the metal model above. That's more or less the geometry we'd have if the sail were not attached to the keel tube at all-- the billow would be highest at the glider centerline, and this would create an obvious, fairly simple anhedral geometry as the sail descended downward toward each wingtip. Tightening the sail would reduce this anhedral geometry. But in reality, we need a theory that recognizes that a billowed sail has dihedral in the inboard portions as well as anhedral in the outboard portions, as illustrated in detail in the related webpage "Notes on sail billow, dihedral, and anhedral in flex-wing hang gliders". We need a theory that recognizes that in the case of an antique hang glider like this one with a very large root chord and tiny wingtip areas, sail billow creates a strong dihedral effect, because the inboard dihedral area is so much larger than the outboard anhedral area. If we simply position the glider in a pitch attitude where the so that the mean chord line is horizontal and then measure the downdroop in the leading-edge tubes, we'll always calculate that the glider has more anhedral with a highly billowed sail than with a taut sail, even in the case of a "standard" Rogallo. We know that that's not really true. The three-dimensional shape of flex-wing hang glider is complex enough that there's really no way to quantify the glider's dihedral or anhedral geometry as "seen" by the airflow or the relative wind, based solely on a single measurement of the leading-geometry or the quarter-chord geometry or the "height" of the wing root relative to the wingtips. Any such measurement will fall short, regardless of whether we specify that the glider's mean chord line be held horizontal during the measurement, or we adopt some other reference system.

How can we use the "theory of swept hinge lines" to better describe the billowed shape of a flex-wing hang glider?

An old Rogallo "standard" hang glider is very simple in shape-- the keel and the leading-edge tubes are equal in length, and the sail fabric is attached to the full length of the keel tube as well as the leading-edge tubes. The left and right sides of the sail each form a simple cone, making a "bi-conical" wing. The nose angle is often less than 90 degrees. As the sail rises upward from the keel tube, it forms a dihedral shape in the inboard part of the wing, and as the sail descends back down to the leading-edge tube, it forms an anhedral shape in the outboard part of the wing. See this figure from Michael Markowski's "The Hang Glider's Bible" (1977). Where are the "hinge lines" of this glider? As a thought experiment, imagine that we outfit the glider with an elastic sail that is initially stretched tight and flat between the keel tubes and leading edges. As we pull the sail up into a billowed shape, it stretches in the middle, and it bends or rotates upward around hinge lines that co-incide with the keel tube and the leading-edge tubes. The metal tubes are the "hinge lines" of this glider. The keel tube can be viewed as a forward-swept hinge line with a sweep angle of 90 degrees, so the sail describes a strong dihedral geometry as it bends up away from this hinge line. This is the dominant geometry of the inboard parts of the wings. The leading-edge tubes are strongly swept aft, so the sail describes a strong anhedral geometry as it bends up away from this hinge line-- this is the dominant geometry of the outboard parts of the wings.

An alternative way to do the same thought experiment is to imagine that we are constructing a standard "Rogallo" hang glider and we haven't yet attached the crossbar in place. We can pull the leading-edge tubes forward until the fabric sail is tight. Then we can move the leading edge tubes aft a few degrees, allowing the sail to billow upward into a pair of cones. This will create both billow and washout. This is exactly how the "standard" Rogallo hang gliders were often constructed-- the sail had a prescribed amount of billow, often 3 1/2 degrees. This was created by adjusting the leading-edge sweep so that the sail was taut, and then increasing the sweep (decreasing the nose angle) by the prescribed amount, e.g. 3.5 degrees. Then the crossbar was bolted into place.

In the old Rogallo "standard" hang glider, sail billow is intimately tied to washout. The larger the "cones" described by the billowed sail, the more washout we'll see when we cut a cross-section of the cone out near the wingtip, cutting parallel to the keel. The upper graph on this page from Michael Markowski's "The Hang Glider's Bible" (1977) shows this relationship.

Now let's look at a modern hang glider. The aspect ratio-- the ratio of span to average chord-- is much higher than with the old Rogallo. The modern glider has a much smaller sweep angle, and a much larger nose angle. Battens define the airfoil shape. The sail is not attached to the keel tube except through the keel pocket behind the hang point-- this keel pocket may be either a tight sleeve or a larger pocket. It's really not accurate to say that sail billow is pulling the sail fabric upward from the keel any more, particularly the forward parts of the keel. With a modern hang glider, it's perhaps more accurate to view sail billow as making the sail "fold" upward along hinge lines shaped like this. See the wing on the right side of the photo (the "aircraft's" left) in particular-- here we've depicted the wing tip chord line as having some freedom to change. This is what we typically see in modern hang gliders, whether the tip construction is a "roach tip" (Wings Will Falcon etc) or a curved tip wand. The wing on the left side of the photo is more representative of a wing with a defined, completely rigid tip strut that keeps the wingtip chord line completely constant-- this construction is not common in modern hang gliders. The inboard "hinge line" is swept forward and so creates dihedral in the inboard part of the wing-- but less so than if the sail were simply billowing up from the keel tube on both sides, as in the old Rogallo "standard". The outboard "hinge line" is swept aft and so creates anhedral in the outboard part of the wing-- but less so than if the leading-edge tube had much more sweep, as in the old Rogallo "standard". There's still a vestige of the old "bi-conical" shape in the modern hang glider, but it's only a faint echo. And the modern hang glider has so much more area in the outboard parts of the sail than did the old Rogallo "standard", that the sail billow creates very different results-- the outboard anhedral areas contribute more roll torque than the inboard dihedral areas. Sail billow has a net dihedral influence on the roll dynamics of an old Rogallo "standard", and a net anhedral influence on the roll dynamics of a modern hang glider. For more, see the discussion of the observed effects of the cam VG system in the related webpage "Notes on sail billow, dihedral, and anhedral in flex-wing hang gliders". But even in a modern hang glider, we can still view sail billow as being created by the fact that the leading-edges of the wing aren't pulled forward as far as the sail fabric will allow. Pulling the leading-edge tubes forward by tensioning a VG system will reduce billow, and will also reduce washout.

It's interesting to consider what controls washout in the outer parts of the wings of a modern hang glider. The relationship is more complex than in the old "standard" Rogallo-- washout can be adjusted independently of billow, to some degree. The trailing edges are somewhat free to "float" up and down at the wingtips in response to changing aerodynamic loads. The battens create a series of "load cells" that transfer some of the rigidity of the keel tube to outboard parts of the wings. For example, if we remove the battens one by one from the wing of a fully assembled hang glider, working from the root toward the tip, we'll see that each missing batten allows the wingtip airfoil more freedom to wash out to a lower angle-of-incidence or angle-of-attack when we pull upward on the trailing-edge of the sail. This is easiest to see in a single-surface glider like a Wills Wing Falcon. Loosening the sail by decreasing the sweep angle of the leading-edge tubes-- e.g. de-tensioning the VG on a glider so equipped-- increases billow in the mid-span parts of the sail and also gives the wingtips more freedom to wash out to a lower angle-of-incidence or angle-of-attack. If the wingtips were held forced to stay in a completely fixed orientation by rigid tip tubes, as suggested by the hinge lines drawn on the wing on the left side of this photo (on the "aircraft's" right side), then sail billow would create an even stronger geometry in the outboard parts of the wings. Compare the two hinge lines in the outboard parts of the wing in the photo above. The hinge line on the left side of the photo has more sweep, and thus creates a stronger anhedral geometry, than the hinge line on the right side of the photo. Another way to think about this to simply look at a hang glider from the side-- we see the anhedral geometry in the outboard part of the wing here, and it's pretty clear that this geometry would be even stronger if we could pull the trailing edge of the tip strut in this photo downward to be more parallel to the keel tube. So really it's most accurate to say that sail billow, not washout, is contributing to anhedral geometry of the outboard parts of the wings. Washout is having the opposite effect-- taking away some of the anhedral geometry that would otherwise be created by the billowed shape of the trailing-edge of the sail.

How would tensioning or de-tensioning a VG affect a glider's roll trim if the wingtip chord lines were fixed by means of rigid tip struts that had no freedom to rotate? As we de-tensioned the VG, the increasing sail billow would still increase the washout angle of the mid-span parts of the wing, but not at the very tips. As suggested above, it appears that this would allow sail billow to create a stronger anhedral geometry in the outboard parts of the wings. With this type of construction, it seems likely that de-tensioning a VG system would actually create a noticeable rolling-in tendency, at least in the case of a cam VG. For any given bank angle and airspeed, the glider might need more high-siding, or less low-siding. On the other hand, if we fit a pulley VG system to a glider with rigidly fixed wingtips, changes in VG setting might have closer to zero effect on the glider's roll trim, because as we de-tensioned the VG, we'd be decreasing "airframe anhedral" at the same time as we increased the anhedral geometry created by sail billow.

On a modern hang glider of conventional wingtip construction-- either "roach tips" or curved tip wands-- what happens if loosen the batten strings so that the wing has more freedom to twist, especially at the wingtip areas? At first glance, it's tempting to view the whole wing as rotating around the leading-edge tube, as suggested by photos 1 through 3 of the metal models above. But twisting a wing isn't the same as rotating the whole wing panel around a hinge line. The geometry described in the last two paragraphs above suggests that if we give the wing more freedom to twist and wash out at the wingtips, we'll take away some of the anhedral geometry that would otherwise be created by sail billow in the outboard parts of the wing.

And what if we "load up" a wing, either by "pulling G's" in maneuvering flight, or by increasing the pilot's weight? Wingtip washout will certainly increase to some degree, as the leading-edge tubes twist and the wing structure flexes. Billow will also increase, as the leading-edge tube bows further aft under load. It would be interesting to view these changes in flight, with a camera mounted several feet outboard of a glider's wingtip, back near the trailing edge. Some anecdotal reports from pilots flying well above the recommended "hook-in" weight for a given glider suggest a net loss of anhedral geometry under load-- in some cases the glider seems to become harder to maneuver in the roll axis, despite the increased weight of the pilot. Obviously this will be highly dependent upon how a specific airframe flexes under load.

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